We consider a G-function F (z) = ∞ k=0 A k z k ∈ K z , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K(z) [d/dz], and a parameter β ∈ Q \ Z 0 . We define a family of G-functions F [s] β,n (z) = ∞ k=0 A k (k + β + n) s z k+n indexed by the integers s and n. Fix α ∈ K * ∩ D(0, R). Let Φ α,β,S be the K-vector space generated by the values F [s] β,n (α), n ∈ N, 0 s S. We show that there exist some positive constants u K,F,β and v F,β such that u K,F,β log(S) dim K Φ α,β,S v F,β S. This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case β = 0. Our proof is an adaptation of their article [6], making use of the André-Chudnovsky-Katz Theorem on the structure of the G-operators and of the saddle point method. Moreover, we rely on Dwork and André's quantitative results on the size of G-operators to obtain an explicit formula for the constant u K,F,β , which was not given in [6] in the case β = 0. This family of special functions has been studied together with the family of E -functions, which are the functions f (z) = ∞ k=0 a k k! z k satisfying a) and such that the a k satisfy the conditions b) and c), since Siegel defined them in 1929 [15]. The most basic example of G-function, which gives it its name, is the geometric series f(−1) k z 2k+1 2k + 1 and the family of hypergeometric functions with rational parameters: if α = (α 1 , .

G-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's G-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a p-adic quantity, the size. Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of a Goperator and certain computable constants depending among others on its solutions. First, we recall André's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of André's generalization of Chudnovsky's Theorem: for f (z) = α,k, c α,k, z α log(z) k f α,k, (z), where f α,k, (z) are G-functions, we can determine an upper bound on the size of the minimal operator L over Q(z) of f (z) in terms of quantities depending on the f α,k, (z) and the rationals α. We give two applications of this result: we estimate the size of a product of two G-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author. Remarks and notations• In this paper, we will call "minimal operator of f over Q(z)", or "minimal operator" where there is no possible ambuigity, any nonzero operator L ∈ Q(z) [d/dz] such that L( f (z)) = 0, and whose order is minimal for f .• For u 1 , . . . , u n ∈ Q, we denote by d (u 1 , . . . , u n ) the denominator of u 1 , . . . , u n , that is to say the smallest d ∈ N * such that d u 1 , . . . , d u n are algebraic integers.• If K is a subfield of Q, we denote by O K the set of algebraic integers of K.

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